1. Evaluate these integrals. (Some of these are indefinite integrals while some are definite integrals. Recall what it means to “evaluate” an indefinite integral versus a definite integral, i.e. what should the final result be?) \[ \int\limits_{1}^{6} \sqrt{3t-2} \,\mathrm{d}t \qquad \qquad \int\limits x\sin\bigl(x^2 + 25\bigr) \,\mathrm{d}x \qquad \qquad \int\limits_{0}^{2} \frac{x^9}{x^{10}+1} \,\mathrm{d}x \qquad \qquad \int (7-2x)\mathrm{e}^{x} \,\mathrm{d}x \qquad \qquad % \int_{2}^{11} \frac{1}{\sqrt{x-2}} \,\mathrm{d}x % \qquad \qquad \int x\ln(x) \,\mathrm{d}x \]
2. What is the total (non-signed) area of the region bound between the graph of \(f(x) = x\ln(x)\) and the \(x\)-axis?
3. A car is driving eastward on I-70 Freeway away from Grand Junction. Its speed, in miles-per-hour, is given by the function \({f(t)= 70+t\cos\bigl(\tfrac{1}{2}t^2\bigr) }\) where \(t\) is the number of hours since it left GJ. How far does it travel in the first four hours? What is the average speed of the car in those first four hours?
4. What is the area of the region bound by the graph of the function \(\displaystyle f(x) = \frac{x-1}{x^6}\) and the positive \(x\)-axis?
5. The function \(\frac{1}{\ln(x)}\) doesn’t have a nice antiderivative, and so the integral \(\int_{2}^{3} \frac{1}{\ln(x)}\,\mathrm{d}x\) can’t be evaluated using the fundamental theorem of calculus. We can however approximate the value of that integral numerically, as a sum of the areas of rectangles that together approximate the region under the curve \(y = \frac{1}{\ln(x)}.\) Approximate the value of this integral using both a left-endpoint sum and a right-endpoint sum.